Thanks, Martini for the clarification. It improves your idea, but I remain to be convinced.
Bear in mind that in your system, if the popularly elected person has to come from the party in government, then in reality ordinay processes of party discipline would result in the party in government choosing its candidate in advance of the election and only putting forward one candidate, or at best one real candidate and a gestural candidate with no chance of winning. This is to avoid the disunity of people within the same party genuinely competing with each other, badmouthing each other, etc. Your modified system will always result in exactly the same outcome as would arise under the present system, but with the expense and delay of a public election. But let’s leave that aside and see what other issues emerge.
When you directly elect a leader (as in the US), the leader gets to appoint a Cabinet, and can replace them at will (subject only to a political cost). Under your proposed system, even if the leader got to appoint their own Cabinet, the Cabinet would still have to come from a pool of people not necessarily of the Leader’s choosing. Your deceptively simple system now has the further complication of probably having to create rules for how Cabinet is appointed. At the moment, individual parties in government use different methods depending on their party rules.
Suppose you have the system just as we have now in most respects, except that when the Premier resigns there is a popular election for his replacement. What happens in Cabinet thereafter? Darwinian processes kick in. Group dynamics elevate the “natural” leader (who would have been elected by the Cabinet in the current system) to a position of prominence, in competition with the elected leader, who may only have got the job because he is the slimiest self-promoter, or has the most telegenic haircut, or whatever. What can a popularly elected leader do if Cabinet does not want him, undermines him at every turn, and votes him down on everything?
Cabinet has to work together intimately, and they have to have confidence in their leader. Of course they are all grown-ups and could (probably) cobble up a modus vivendi even if the leader is not their optimal choice, but the risk of instability is the problem.
In the present system, Cabinet kicks the leader out if he is not performing. They are very fine-tuned to public opinion so it ought not be thought that the appointment of a leader will be capriciously contrary to the public’s wishes. And this process if very flexible and quick.
In your proposed system, what happens if the popularly elected leader is a dud (either because he loses the confidence of Cabinet or Parliament)? Parliament cannot just replace him, otherwise they could do it the moment he came into office making the popular election pointless. What then is required is a system of recall (such as was used against Governor Gray in California prior to Gov. Schwarzenegger’s term). This is cumbersome and protracted. Again, a deceptively simple idea breeds unforseen complications.
I well understand why it might seem attractive to have arrangements for a popularly elected leader somehow fitted within our present system. For mine, though, the sense of satisfaction derived from having a direct say in the leader is illusory. Governments do not keep leaders for long who are albatrosses around their necks in the public eye. Look at all the recent NSW Premiers. Parties vote in as leaders the people who can win them elections, that is, the leaders the public wants.
In short, I can’t see a satisfactory way of having both a popularly elected leader and a Parliamentary system that is remotely similar to the present. (I am talking about the direct election of a Premier or Prime Minister here.) Any attempt to hybridise the Washington model and the Westminster model will necessarily struggle.